Triples and quadruples of consecutive squares or non-squares in a finite field

Let [Formula: see text] be the finite field of odd prime power order [Formula: see text]. We find explicit expressions for the number of triples [Formula: see text] of consecutive non-zero squares in [Formula: see text] and similarly for the number of triples of consecutive non-square elements. A key ingredient is the evaluation of Jacobsthal sums over general finite fields by Katre and Rajwade. This extends results of Monzingo (1985) to non-prime fields. Curiously, the same machinery allows the evaluation of the number of consecutive quadruples [Formula: see text] of square and non-squares over [Formula: see text], when [Formula: see text] is a power of 5.